Problem 10 - A Graphical Approach to Precalculus with Limits

Question

For each statement, write an equivalent statement in exponential form. Do not use a calculator. $$\log _{5} 5=1$$

Step-by-Step Solution

Answer

The equivalent exponential form is \(5^1 = 5\).

Step 1: Identify the logarithmic statement

We are given the logarithmic statement \( \log_{5} 5 = 1 \). This means that the base of the logarithm is 5, the argument is 5, and the logarithm equals 1.

Step 2: Understand the definition of logarithms

Recall the definition of logarithms: if \( \log_b a = c \), then the equivalent exponential form is \( b^c = a \). This means that a logarithm is the power to which a base number is raised to obtain another number.

Step 3: Apply the definition of logarithms

Using the definition of logarithms, convert \( \log_{5} 5 = 1 \) into its exponential form. By definition, this can be rewritten as \( 5^1 = 5 \). This shows that the number \( 5 \) to the power of \( 1 \) equals \( 5 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are a significant part of mathematics, specifically in algebra. The primary purpose of these functions is to determine the power or exponent that a certain base number must be raised to, in order to yield another number. This process makes logarithms essentially the inverse of exponential functions.
Logarithms follow the basic formula: \( \log_b a = c \), where:

• \( b \) is the base, and it must be a positive number different from 1.
• \( a \) is the argument, the number you are taking the logarithm of, which must be positive.
• \( c \) is the exponent or power.

When you see \( \log_5 5 = 1 \), it means that \( 5 \), the base, raised to what power, gives you \( 5 \).
This statement is powerful for understanding the core concept of logarithms, helping to build a strong mathematical foundation.

Logarithm Properties

Logarithms have several important properties that make them useful for simplifying complex calculations and solving equations. Understanding these can enhance your capability to work with logarithmic and exponential forms.
Here are some key properties of logarithms to remember:

• Product Rule: \( \log_b (xy) = \log_b x + \log_b y \)
• Quotient Rule: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• Power Rule: \( \log_b (x^c) = c \cdot \log_b x \)
• Change of Base Formula: \( \log_b a = \frac{\log_k a}{\log_k b} \) for any positive base \( k \).

These properties allow us to manipulate and simplify logarithmic expressions with greater ease.
They provide tools for converting between multiplication, division, and exponentiation within logarithmic expressions, both deepening understanding and enhancing problem-solving capabilities.

Conversion Between Forms

The conversion between logarithmic and exponential forms is a fundamental skill. It's commonly used not only in academics but also in practical applications involving exponential growth or decay scenarios.
Understanding the interplay between these forms involves recognizing that logarithms and exponents are inverse processes.
Here is how this works: for the logarithmic expression \( \log_b a = c \), the equivalent exponential format is \( b^c = a \).
Let's consider the example \( \log_5 5 = 1 \):

• Logarithmic Form: Indicates \( 5 \), as the base, is raised to the power of \( 1 \), giving the number \( 5 \).
• Exponential Form: Written as \( 5^1 = 5 \).

This process of conversion helps in visualizing and solving problems where you need to switch between these perspectives.
Mastering this skill facilitates understanding of more complex concepts like logarithmic scales, and can significantly aid in solving various mathematical problems effortlessly.